1. Field of the Invention
The invention relates to a network of the neural cellular type useful to implement a so-called Chua's circuit. More particularly, the invention relates to a cellular neural network of the type that includes at least first, second and third cells, each having respective first and second input terminals and respective state terminals, wherein the first and second input terminals are to receive first and second reference signals, respectively.
The invention concerns in particular, though not exclusively, a cellular neural network adapted to implement a Chua's circuit of the so-called unfolded type, and this description will make reference to that field of application for simplicity of illustration.
2. Discussion of the Related Art
Cellular neural networks are a new class of non-linear analog circuits developed by Chua and Yang in 1988.
As those skilled in the art know well, Chua's circuit is an oscillatory circuit which exhibits a large variety of fork and chaos phenomena.
Shown diagrammatically in FIG. 1 is an exemplary Chua's circuit, generally designated 1, which comprises first C1, second C2 and third L linear energy-storage elements. These elements are connected, in parallel with one another, across an additional non-linear element Nr which is further connected to a potential reference, such as a signal ground GND.
In particular, the additional non-linear element Nr is a non-linear resistor commonly referred to as "Chua's diode".
Furthermore, each of the first C1 and second C2 linear elements has a first terminal connected to the ground GND, and a second terminal connected together through a linear element R, specifically a resistor.
The equations of state for the circuit 1 are the following: ##EQU1## where, .nu..sub.1 and .nu..sub.2 are a first and a second voltage value appearing across the first C1 and second C2 linear elements, respectively;
i.sub.3 is the value of a current flowing through the third linear element L; PA1 G is the conductance of the linear element R, i.e., G=1/R; and PA1 g(.nu..sub.1) is a segmental linear function described by the following relation: ##EQU2## in which, G.sub.a and G.sub.b are a first and a second slope of the function curve; and PA1 B.sub.p is a point of inflection on the function curve. PA1 x, y, z are the state variables of the dimensionless equations system (3.1), (3.2), (3.3), (3.4); and PA1 .alpha., .beta., .gamma., m.sub.0, m.sub.1 are the parameters of the dimensionless equations system. PA1 A(ij;k,l) and B(ij;k,l) are constant coefficients commonly referred to as "cloning templates"; and PA1 N.sub.r is the neighborhood of the cell C(k,l) as described by the following relation: EQU N.sub.r (i,j)={C(k,l).vertline.max(.vertline.k-i.vertline., .vertline.l-j.vertline.).vertline..ltoreq.r, 1.ltoreq.k.ltoreq.M; 1.ltoreq.l.ltoreq.N} (6.4) PA1 j is the cell index; PA1 x.sub.j is the state variable; PA1 a.sub.j is a constant parameter; PA1 i.sub.j is a threshold value; PA1 G.sub.0 and G.sub.s are linear combinations of the outputs and the state variables of cells connected to the cell considered; and PA1 y.sub.j is the cell output as per the following relation: EQU y.sub.j =0.5[.vertline.x.sub.j +1.vertline.-.vertline.x.sub.j -1.vertline.](9) PA1 y.sub.1, y.sub.2, y.sub.3 are the corresponding outputs. PA1 By arranging for the parameters a.sub.ij and s.sub.ij of the system to satisfy the following relations, EQU a.sub.12 =a.sub.13 =a.sub.2 =a.sub.23 =a.sub.3 =a.sub.21 =a.sub.31 =0(11.1) EQU s.sub.13 =s.sub.31 =s.sub.22 =0 (11.2) EQU i.sub.1 =i.sub.2 =i.sub.3 =0 (11.1) PA1 I71 opposite of a first reference signal V1; PA1 I72 opposite of a second reference signal V2; PA1 O7 opposite of a first output signal y1; PA1 T7 opposite of a first state signal x1; PA1 I81 opposite of the first reference signal V1; PA1 I82 opposite of the second reference signal V2; PA1 O8 a second state signal x2; PA1 T8 opposite of the second state signal x2; PA1 I91 opposite of the first reference signal V1; PA1 I92 opposite of the second reference signal V2; PA1 O9 opposite of an output signal y3; PA1 T9 opposite of a third state signal x3.
Shown in FIG. 2 is a possible plot of the function g(.nu..sub.1) versus voltage .nu..sub.1. It can be seen from this plot that the additional non-linear element Nr is a voltage-controlled element, since the current flowing therethrough is a function of the voltage applied to its terminals.
A solution of the equations of state (1.1), (1.2), (1.3) above has been simulated to show that Chua's circuit 1 has an attractor DS, known as a double-scroll attractor. In addition, a variety of forks and chaotic attractors can be obtained. The circuit 1 can be regarded as a basic element of chaotic systems.
It would also be possible to introduce a further linear element R.sub.0, in series with the third linear element L. The resulting circuit is referred to as an "unfolded" Chua's circuit, and is described by the following equations of state: ##EQU3## where, .function.(.nu..sub.1) is a segmental linear function described by the following relation: ##EQU4##
The equations of state (2.1), (2.2), (2.3) of the unfolded Chua's circuit can also be written in the following dimensionless forms: EQU x=.alpha.[-h(x)] (3.1) EQU y=x-y+z (3.2) EQU z=-.beta.y-yz (3.3)
with: EQU h(x)=m.sub.1 x+0.5(m.sub.0 -m.sub.1) [.vertline.x+1.vertline.-.vertline.x-1.vertline.] (3.4)
where,
By a simple calculation process, the following relations can be found for the parameters involved:
______________________________________ x = v.sub.1 /E y = v.sub.2 /E z = i.sub.3 /(EG) (4.1) t =(.tau.G)/C.sub.2 m.sub.0 = (G.sub.a /G) + 1 m.sub.1 = (G.sub.b /G) + 1 (4.2) .alpha. = C.sub.2 /C.sub.1 .beta. = C.sub.2 /(LG.sup.2) .gamma. = (C.sub.2 /R.sub.0)/(GL) (4.3) ______________________________________
Chua's circuit 1 is formed, specifically in its unfolded configuration, in a known manner from cells of Cellular Neural Networks (CNNs), as described in an article, "Chua's Circuit Can Be Generated by CNN Cells" by Arena, Baglio, Fortuna and Manganaro, published in IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS--PART I, Feb. 2, 1995, which is hereby incorporated by reference.
Cellular neural networks are a new class of non-linear analog circuits introduced in 1988 by Chua and Yang. FIG. 3 is a typical diagram for a basic cell 2 in cellular neural networks or CNNs.
This basic cell 2 has an input voltage generator Eij connected to an input terminal IN which is to receive an input signal u.sub.ij.
Also provided is a first divider 3 which has a bias current generator I, a resistive element Rx, a capacitive element C, and first Ixu and second Ixy current generators which are voltage-controlled in parallel with each other. The basic cell 2 further includes a second divider 4 having a third voltage-controlled current generator Iyx in parallel with a resistive element Ry, and an output terminal OUT of the basic cell 2.
The output terminal OUT supplies an output voltage y.sub.ij, and a voltage corresponding to a state variable x.sub.ij of the cell 2 is present on the first divider 3. The output signal y.sub.ij is tied to the state variable x.sub.ij by the following non-linear equation: EQU y.sub.ij =0.5[.vertline.x.sub.ij +1.vertline.-.vertline.x.sub.ij -1.vertline.] (5)
A cellular neural network or CNN includes a plurality of basic cells 2 interconnected with adjacent cells, the latter forming the cell "neighborhood".
A simple implementation of a CNN includes a bi-dimensional array of M.times.N identical cells. An array 5 of this kind is shown in FIG. 4, where the nomenclature C(i,j) indicates a cell located in the i-th row and j-th column of the array 5, with i varying between 1 and N, and j varying between 1 and M.
Each cell C(i,j) interacts with its neighbors through voltage-controlled current generators, as described by the following relations: EQU I.sub.xy (i,j;k,l)=A(i,j;k,l)y.sub.kl (6.1) EQU I.sub.xu (i,j;k,l)=B(i,j;k,l)u.sub.kl (6.2) EQU C(k,l).epsilon.N.sub.r (i,j) (6.3)
where,
From the previous equations of state (6.1), (6.2) and (6.3), applicable to a single cell C(k,l), a relation can be obtained which applies to all the cells in a cellular neural network (CNN). Thus the following equation of state is arrived at: ##EQU5##
Therefore, a cellular neural network (CNN) is an n-dimensional array of dynamic systems, called the cells, which satisfy two conditions. First, most of the interactions are local and have a finite radius.
Second, all the state variables are signals which vary continuously.
A basic cell in a cellular neural network (CNN) can be modelled as described by the following non-linear equation of state: EQU x.sub.j =-x.sub.j +a.sub.j y.sub.j +G.sub.0 +G.sub.s +i.sub.j(8)
where,
In the simplified instance of three cells being connected, the model of the dynamic system represented by the equation of state (7) becomes, ##EQU6## where, x.sub.1, x.sub.2, x.sub.3 are the state variables of the system; and
the system model can be simplified to the following form: EQU x.sub.1 =-x.sub.1 +a.sub.1 y.sub.1 +s.sub.11 x.sub.1 +s.sub.12 x.sub.2(12.1 ) EQU x.sub.2 =-x.sub.2 +s.sub.21 x.sub.1 +s.sub.23 x.sub.3 (12.2) EQU x.sub.3 =-x.sub.3 +s.sub.32 x.sub.2 +s.sub.33 x.sub.3 (12.3)
The equations system (12.1), (12.2) and (12.3) correspond to the linear equations system (2.1), (2.2) and (2.3) which model an unfolded Chua's circuit.
Therefore, it can be concluded that an unfolded Chua's circuit can be implemented by a circuit using a CNN made up of three basic cells.
Shown diagrammatically in FIG. 5 is a cellular neural network 6 which includes first 7, second 8 and third 9 basic cells.
Each of the basic cells 7, 8, 9 has a pair of input terminals, an output terminal and a state terminal. In particular, the first basic cell 7 has a first input terminal I71 connected to an output terminal O7, a second input terminal I72 connected to a state terminal T8 of the second basic cell 8, and a state terminal T7 connected to a first input terminal I81 of the second basic cell 8.
The second basic cell 8 has an output terminal O8 and a second input terminal I82 which are respectively connected to first I91 and second I92 input terminals of the third basic cell 9.
Furthermore, the third basic cell 9 has a state terminal T9 connected to the first input terminal I92, and an output terminal O9 which forms an output terminal OUT of the network 6.
Present on the terminals of the basic cells 7, 8, 9 are the following signals:
Each basic cell 7, 8, 9 can be implemented by a circuit according to the diagram shown in FIG. 6, which illustrates a generic basic cell BC.
The cell BC has first I1 and second I2 input terminals which receive the opposite of first V1 and second V2 reference signals, respectively.
The first input terminal I1 is connected, through a first linear element R1, which may be a resistor, to a first inverting input terminal I3 of a first operational amplifier A1 which has a second non-inverting input terminal I4 connected to a potential reference, such as a ground reference GND, and an output terminal O1 connected to the second input terminal I2 through a series of a second R2 and third R3 linear elements, e.g. resistors.
The output terminal O1 of the first operational amplifier A1 is further connected to an internal circuit node X through a fourth linear element R4.
This internal circuit node X is further connected to the ground reference GND, through a linear element Cj, such as a capacitor, and to a first inverting input terminal I5 of a second operational amplifier A2 through a fifth linear element R5.
The voltage present across the linear element Cj forms a state signal x.sub.j of the basic cell BC.
The second operational amplifier A2 has a second non-inverting input terminal 16 connected to the ground reference GND, and an output terminal O2 connected to the aforesaid first inverting input terminal I5 through a sixth linear element R6.
The output terminal O2 of the second operational amplifier A2 forms a state terminal T of the basic cell BC. The opposite of the state signal x.sub.j is present on the state terminal T.
Finally, the internal circuit node X is connected to a first inverting input terminal I7 of a third operational amplifier A3 through a seventh linear element R7, e.g. a resistor.
The third operational amplifier A3 has a second non-inverting input terminal I8 connected to the ground reference GND, and an output terminal O3 connected to the first input terminal through an eighth linear element R8, e.g. a resistor.
The output terminal O3 of the third operational amplifier A3 is connected to an output terminal Y of the basic cell BC through a ninth linear clement R9, e.g. a resistor.
The output terminal Y of the basic cell BC is further connected to ground reference the GND through a tenth linear element R10, e.g. a resistor.
The opposite of an output signal y.sub.j is present on the output terminal Y of the basic cell BC.
The underlying technical problem of this invention is to provide a programmable cellular neural network which has such constructional and functional features as to represent an improvement on similar prior networks.